If I am given a set of points in $\mathbb{R}^3$ and asked if they are collinear or not, can I test the set of points for linear independence to see if they lie on the same line or not? Then I would form the $3\times 3$ matrix and take the determinant. If the determinant is $0$, the set is linearly dependent and they are collinear points. If the $\det\neq0$, then the points are linearly independent and hence not collinear.
My question is, is this method valid to determine if a set of points in $\mathbb{R}^3$ lie on the same straight line or not?
No, it doesn't work. If it worked, then any three points of the form $(a,b,0)$ would be collinear, since the determinant that you mentioned would be $0$.
It order to determin whether or not $A$, $B$ and $C$ are collinear, comput $C-A$ and $B-A$. The points will be collinear if and only if $C-A=\lambda(B-A)$ for some $\lambda\in\mathbb R$.